
=======
 woOW!
=======

 "Tiny stubs of an endless curiosity... potentially useful stubs of that"

#0001, Today featuring the differences of the form:

A243106(n+1,y)-( A243106(n+1,y)\A243106(n,y) )*A243106(n,y)

(where "\" is the integer division, "div" or "DIV" in some environments)

More precisely: Their digits in base y when evaluated at x=y;

(Note: On the present time, A243106 still being at draft stage, proposed).

--------------------[ View in order to believe (Begin) ]--------------------

Reading GPRC: /etc/gprc ...Done.

                                          GP/PARI CALCULATOR Version 2.7.1 (released)
                                   i686 running linux (ix86/GMP-5.0.5 kernel) 32-bit version
                                        compiled: Aug 19 2014, gcc version 4.7.2 (GCC) 
                                                    threading engine: single
                                         (readline v5.2 enabled, extended help enabled)

                                             Copyright (C) 2000-2014 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 966367616, primelimit = 100000000
? ap(n, x)={my(s); forprime(p=1, n, s+=x^p); s}
time = 1 ms.
%1 = (n,x)->my(s);forprime(p=1,n,s+=x^p);s
? a=(n, x=10)->(x^(n+1)-1)/(x-1)-2*ap(n, x)-1
%2 = (n,x=10)->(x^(n+1)-1)/(x-1)-2*ap(n,x)-1
? woOW=(m,x)->vector(m,j,digits(a(j+1,x)-(a(j+1,x)\a(j,x))*a(j,x),x))
%3 = (m,x)->vector(m,j,digits(a(j+1,x)-(a(j+1,x)\a(j,x))*a(j,x),x))
? woOW(10,3)
time = 2 ms.
%4 = [[], [1, 0], [1, 2, 0], [1, 2, 0, 0], [1, 2, 1, 0, 0], [1, 2, 2, 2, 2, 0], [2, 0, 1, 2, 1, 0, 0], [1, 2, 2, 0], [1, 1, 1, 1, 1, 2, 1, 1, 0], [1, 0, 2, 0, 2, 0, 1, 2, 1, 0, 0]]
? woOW(10,8)
time = 2 ms.
%5 = [[], [6, 0], [1, 7, 0], [4, 7, 2, 0], [6, 7, 1, 0, 0], [7, 0, 4, 7, 2, 0], [7, 0, 6, 7, 1, 0, 0], [1, 7, 7, 0], [1, 6, 1, 6, 1, 7, 6, 1, 0], [1, 0, 7, 0, 7, 0, 6, 7, 1, 0, 0]]
? woOW(10,10)
time = 1 ms.
%6 = [[], [8, 0], [1, 9, 0], [6, 9, 2, 0], [8, 9, 1, 0, 0], [9, 0, 6, 9, 2, 0], [9, 0, 8, 9, 1, 0, 0], [1, 9, 9, 0], [1, 8, 1, 8, 1, 9, 8, 1, 0], [1, 0, 9, 0, 9, 0, 8, 9, 1, 0, 0]]
? woOW(10,12)
time = 2 ms.
%7 = [[], [10, 0], [1, 11, 0], [8, 11, 2, 0], [10, 11, 1, 0, 0], [11, 0, 8, 11, 2, 0], [11, 0, 10, 11, 1, 0, 0], [1, 11, 11, 0], [1, 10, 1, 10, 1, 11, 10, 1, 0], [1, 0, 11, 0, 11, 0, 10, 11, 1, 0, 0]]
? woOW(10,16)
time = 1 ms.
%8 = [[], [14, 0], [1, 15, 0], [12, 15, 2, 0], [14, 15, 1, 0, 0], [15, 0, 12, 15, 2, 0], [15, 0, 14, 15, 1, 0, 0], [1, 15, 15, 0], [1, 14, 1, 14, 1, 15, 14, 1, 0], [1, 0, 15, 0, 15, 0, 14, 15, 1, 0, 0]]
? woOW(10,36)
time = 2 ms.
%9 = [[], [34, 0], [1, 35, 0], [32, 35, 2, 0], [34, 35, 1, 0, 0], [35, 0, 32, 35, 2, 0], [35, 0, 34, 35, 1, 0, 0], [1, 35, 35, 0], [1, 34, 1, 34, 1, 35, 34, 1, 0], [1, 0, 35, 0, 35, 0, 34, 35, 1, 0, 0]]
? woOW(10,2) /*
              * Binary, too limited as alphabet, at least in order to behave alike the greater radices
			  */
time = 2 ms.
%10 = [[], [], [1, 1, 0], [1, 0, 0], [1, 1, 0, 0], [1, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0, 0], [1, 1, 1, 0], [1, 0, 1, 0, 1, 1, 0, 1, 0], [1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0]]
? \q
Goodbye!!

--------------------[ View in order to believe (End) ]--------------------


===========================
 Where is the woOW?
 What is it consisting of?
===========================

 This is it: By observing in detail such sample outputs: We are in presence
 of the same kind of uniformly-structured behavior already described for
 A243106 itself when considered from a more general viewpoint in other
 bases.

 But... Why?.
 
 Also: There is implicitly an open question: Does this same particular form of
 "first differences" generate the same kind of results when applied
 to A215940?

 Briefly reviewed, the first 50 terms of this "woOW!" for decimal or base 10, are:
 
 0, 80, 190, 6920, 89100, 906920, 9089100, 1990, 181819810, 10909089100,
 21999998010, 21999998010, 21999998010, 90889090910900, 181800181819810,
 10909110909089100, 22000021999998010, 22000021999998010, 22000021999998010,
 90889090889090910900, 181800181800181819810, 10909110909110909089100,
 22000022000021999998010, 889090889090889090910900, 1800181800181800181819810,
 1800181800181800181819810, 1800181800181800181819810, 11109110909110909110909089100,
 22220022000022000021999998010, 22220022000022000021999998010, 22220022000022000021999998010,
 90888890889090889090889090910900, 181800001800181800181800181819810, 181800001800181800181800181819810,
 181800001800181800181800181819810, 1110909111109110909110909110909089100, 2222000022220022000022000021999998010,
 88889090888890889090889090889090910900, 180000181800001800181800181800181819810, 10911110909111109110909110909110909089100,
 22002222000022220022000022000021999998010, 22002222000022220022000022000021999998010, 22002222000022220022000022000021999998010,
 90889088889090888890889090889090889090910900, 181800180000181800001800181800181800181819810, 10909110911110909111109110909110909110909089100,
 22000022002222000022220022000022000021999998010, 889090889088889090888890889090889090889090910900, 1800181800180000181800001800181800181800181819810,
 1800181800180000181800001800181800181800181819810
 
 ...That's all for now. Best regards.
 
------------------------------------------------
   woOW! #0001 By: R. J. Cano; On Aug 19 2014.
   Edited @ 1024*600 full screen using Linux
   & "SciTE 2.25 by Neil Hodgson" as editor.
------------------------------------------------

(EOF)